Subgroups — C₅×#5634 #11756019

Order 960.

484 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#946	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#897	C₂	1	2	yes	maximal, complemented
6	C₅×#896	C₂	1	2	yes	maximal, complemented
7	C₅×#896	C₂	1	2	yes	maximal, complemented
8	C₅×D₃×C₂²⋊C₄	C₂	1	2	yes	maximal
9	C₅×#878	C₂	1	2	yes	maximal, complemented
10	C₅×#543	—	3	1	no	maximal, Hall
11	C₂³×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	2	yes	complemented
18	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₅×#217	C₂²	1	2	yes	—
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	2	yes	complemented
28	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
31	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
32	#5634	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×C₂²⋊Q₈	—	3	2	no	—
34	C₅×C₂²⋊Q₈	—	3	1	no	—
35	C₅×C₂²⋊Q₈	—	3	2	no	—
36	C₅×C₂²⋊Q₈	—	3	2	no	—
37	C₅×C₂²⋊Q₈	—	3	2	no	—
38	C₅×C₂²⋊Q₈	—	3	2	no	—
39	C₅×C₂²⋊Q₈	D₃	1	1	yes	characteristic, complemented
40	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
41	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂²×C₅×D₃	—	4	1	no	—
44	C₂²×C₅×D₃	—	4	1	no	—
45	C₂²×C₅×D₃	Q₈	1	4	yes	complemented
46	C₂²×C₅×D₃	C₂³	1	2	yes	—
47	C₃×C₅×Q₈	—	4	1	no	—
48	C₂×C₃×C₄×C₅	—	4	1	no	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₅×Dic₃	—	4	1	no	—
54	C₂×C₅×Dic₃	C₂³	1	2	yes	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₄×C₅×D₃	—	4	2	no	—
58	C₄×C₅×D₃	—	4	2	no	—
59	C₅×Dic₆	—	4	1	no	—
60	C₅×Dic₆	—	4	2	no	—
61	C₃×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
62	#946	C₂×C₅	1	2	yes	complemented
63	C₂×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic
64	#924	C₂×C₅	1	1	yes	characteristic, complemented
65	#897	C₂×C₅	1	2	yes	complemented
66	#896	C₂×C₅	1	2	yes	complemented
67	#896	C₂×C₅	1	2	yes	complemented
68	D₃×C₂²⋊C₄	C₂×C₅	1	2	yes	—
69	#878	C₂×C₅	1	2	yes	complemented
70	C₂⁴×C₅	—	3	1	no	—
71	C₂×C₅×Q₈	—	3	1	no	—
72	C₂×C₅×Q₈	—	3	2	no	—
73	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂²×C₄×C₅	—	3	1	no	—
75	C₂²×C₄×C₅	—	3	2	no	—
76	C₂²×C₄×C₅	—	3	2	no	—
77	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	3	2	no	—
79	C₅×C₄⋊C₄	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	1	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
84	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	complemented
85	C₅×C₂²⋊C₄	—	3	2	no	—
86	C₅×C₂²⋊C₄	—	3	2	no	—
87	C₅×C₂²⋊C₄	—	3	2	no	—
88	C₅×C₂²⋊C₄	—	3	2	no	—
89	C₅×C₂²⋊C₄	—	3	2	no	—
90	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
91	#543	—	3	1	no	Sylow
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
95	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×C₅×D₃	—	2	2	no	—
97	C₂×C₅×D₃	—	2	2	no	—
98	C₂×C₅×D₃	—	4	4	no	—
99	C₂×C₅×D₃	C₂×Q₈	1	4	yes	—
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₅×Dic₃	—	2	2	no	—
105	C₅×Dic₃	—	2	2	no	—
106	C₅×Dic₃	—	2	2	no	—
107	C₂³×D₃	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
109	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
110	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
112	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
113	C₂×Dic₆	C₂²×C₅	1	2	yes	complemented
114	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic, complemented
115	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
116	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
117	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
118	#217	C₂²×C₅	1	2	yes	—
119	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
120	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
121	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
122	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
123	C₄⋊Dic₃	C₂²×C₅	1	2	yes	complemented
124	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
125	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
126	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
127	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
128	C₂³×C₅	—	3	2	no	—
129	C₂³×C₅	—	3	4	no	—
130	C₂³×C₅	—	12	1	no	—
131	C₂³×C₅	—	12	1	no	—
132	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₅×Q₈	—	4	1	no	—
134	C₅×Q₈	—	12	1	no	—
135	C₅×Q₈	—	12	2	no	—
136	C₂×C₄×C₅	—	3	2	no	—
137	C₂×C₄×C₅	—	3	2	no	—
138	C₂×C₄×C₅	—	3	2	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	12	2	no	—
141	C₂×C₄×C₅	—	12	2	no	—
142	C₂×C₄×C₅	—	12	1	no	—
143	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
147	C₂²⋊Q₈	—	3	2	no	—
148	C₂²⋊Q₈	—	3	1	no	—
149	C₂²⋊Q₈	—	3	2	no	—
150	C₂²⋊Q₈	—	3	2	no	—
151	C₂²⋊Q₈	—	3	2	no	—
152	C₂²⋊Q₈	—	3	2	no	—
153	C₂²⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
154	C₂×C₂²⋊C₄	—	3	1	no	—
155	C₂×C₂²⋊C₄	—	3	2	no	—
156	C₂×C₃×C₅	—	2	2	no	—
157	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
160	C₅×D₃	—	2	4	no	—
161	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
162	C₂²×D₃	—	4	1	no	—
163	C₂²×D₃	—	4	1	no	—
164	C₂²×D₃	C₅×Q₈	1	4	yes	complemented
165	C₂²×D₃	C₂³×C₅	1	2	yes	—
166	C₃×Q₈	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
170	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
171	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
172	C₂×Dic₃	—	4	1	no	—
173	C₂×Dic₃	C₂³×C₅	1	2	yes	—
174	C₂×Dic₃	C₂³×C₅	1	2	yes	—
175	C₂×Dic₃	C₂³×C₅	1	2	yes	—
176	C₄×D₃	—	4	2	no	—
177	C₄×D₃	—	4	2	no	—
178	Dic₆	—	4	1	no	—
179	Dic₆	—	4	2	no	—
180	C₂²×C₅	—	2	1	no	—
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	3	4	no	—
183	C₂²×C₅	—	6	2	no	—
184	C₂²×C₅	—	6	2	no	—
185	C₂²×C₅	—	12	4	no	—
186	C₂²×C₅	D₃×Q₈	1	2	yes	—
187	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	6	2	no	—
193	C₄×C₅	—	6	2	no	—
194	C₄×C₅	—	6	2	no	—
195	C₂⁴	—	3	1	no	—
196	C₂×Q₈	—	3	1	no	—
197	C₂×Q₈	—	3	2	no	—
198	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	2	no	—
201	C₂²×C₄	—	3	2	no	—
202	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
209	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	complemented
210	C₂²⋊C₄	—	3	2	no	—
211	C₂²⋊C₄	—	3	2	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	—	3	2	no	—
214	C₂²⋊C₄	—	3	2	no	—
215	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
216	C₃×C₅	#543	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	C₂×C₅×Q₈	1	2	yes	—
220	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
221	C₂×D₃	—	2	2	no	—
222	C₂×D₃	—	2	2	no	—
223	C₂×D₃	—	4	4	no	—
224	C₂×D₃	C₂×C₅×Q₈	1	4	yes	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	Dic₃	—	2	2	no	—
230	Dic₃	—	2	2	no	—
231	Dic₃	—	2	2	no	—
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	6	4	no	—
234	C₂×C₅	#1004	1	1	yes	characteristic
235	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
236	C₂×C₅	#1009	1	1	yes	characteristic
237	C₂³	—	3	2	no	—
238	C₂³	—	3	4	no	—
239	C₂³	—	12	1	no	—
240	C₂³	—	12	1	no	—
241	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
242	Q₈	—	4	1	no	—
243	Q₈	—	12	1	no	—
244	Q₈	—	12	2	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	12	2	no	—
250	C₂×C₄	—	12	2	no	—
251	C₂×C₄	—	12	1	no	—
252	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
255	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
258	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
260	D₃	—	2	4	no	—
261	C₅	#5634	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	3	4	no	—
265	C₂²	—	6	2	no	—
266	C₂²	—	6	2	no	—
267	C₂²	—	12	4	no	—
268	C₂²	C₅×D₃×Q₈	1	2	yes	—
269	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	6	2	no	—
275	C₄	—	6	2	no	—
276	C₄	—	6	2	no	—
277	C₃	C₅×#543	1	1	yes	characteristic, complemented, Sylow
278	C₂	—	2	2	no	—
279	C₂	—	6	4	no	—
280	C₂	C₅×#1004	1	1	yes	characteristic
281	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
282	C₂	C₅×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂³×C₅×D₃	C₂²
hypercenter	C₅×C₂²⋊Q₈	D₃
hypocenter	C₃	C₅×#543
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
2. C₅×#946	C₂	semidirect
4. C₅×#924	C₂	semidirect
5. C₅×#897	C₂	semidirect
6. C₅×#896	C₂	semidirect
7. C₅×#896	C₂	semidirect
9. C₅×#878	C₂	semidirect
12. C₂×C₃×C₅×Q₈	C₂²	semidirect
17. C₂×C₅×Dic₆	C₂²	semidirect
18. C₂×C₅×Dic₆	C₂²	semidirect
19. C₃×C₅×C₄⋊C₄	C₂²	semidirect
20. C₃×C₅×C₄⋊C₄	C₂²	semidirect
27. C₅×C₄⋊Dic₃	C₂²	semidirect
28. C₅×C₄⋊Dic₃	C₂²	semidirect
29. C₅×Dic₃⋊C₄	C₂²	semidirect
30. C₅×Dic₃⋊C₄	C₂²	semidirect
31. C₅×Dic₃⋊C₄	C₂²	semidirect
32. #5634	C₅	direct (N × H)
39. C₅×C₂²⋊Q₈	D₃	semidirect
45. C₂²×C₅×D₃	Q₈	semidirect
61. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
62. #946	C₂×C₅	semidirect
64. #924	C₂×C₅	semidirect
65. #897	C₂×C₅	semidirect
66. #896	C₂×C₅	semidirect
67. #896	C₂×C₅	semidirect
69. #878	C₂×C₅	semidirect
73. C₂×C₅×Q₈	C₂×D₃	semidirect
83. C₅×C₄⋊C₄	C₂×D₃	semidirect
84. C₅×C₄⋊C₄	C₂×D₃	semidirect
108. C₂×C₃×Q₈	C₂²×C₅	semidirect
113. C₂×Dic₆	C₂²×C₅	semidirect
114. C₂×Dic₆	C₂²×C₅	semidirect
115. C₃×C₄⋊C₄	C₂²×C₅	semidirect
116. C₃×C₄⋊C₄	C₂²×C₅	semidirect
123. C₄⋊Dic₃	C₂²×C₅	semidirect
124. C₄⋊Dic₃	C₂²×C₅	semidirect
125. Dic₃⋊C₄	C₂²×C₅	semidirect
126. Dic₃⋊C₄	C₂²×C₅	semidirect
127. Dic₃⋊C₄	C₂²×C₅	semidirect
153. C₂²⋊Q₈	C₅×D₃	semidirect
164. C₂²×D₃	C₅×Q₈	semidirect
198. C₂×Q₈	C₂×C₅×D₃	semidirect
208. C₄⋊C₄	C₂×C₅×D₃	semidirect
209. C₄⋊C₄	C₂×C₅×D₃	semidirect
216. C₃×C₅	#543	semidirect
261. C₅	#5634	direct (N × H)
277. C₃	C₅×#543	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×C₂²⋊Q₈	2
C₅×#946	2
C₂×C₅×(C₂×D₃)⋊C₄	2
C₅×#924	2
C₅×#897	2
C₅×#896	2
C₅×D₃×C₂²⋊C₄	2
C₅×#878	2
C₅×#543	3
#5634	5

Subgroups — C5×#5634 #11756019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5634 #11756019